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Glivenko sequent classes in the light of structural proof theory

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Abstract

In 1968, Orevkov presented proofs of conservativity of classical over intuitionistic and minimal predicate logic with equality for seven classes of sequents, what are known as Glivenko classes. The proofs of these results, important in the literature on the constructive content of classical theories, have remained somehow cryptic. In this paper, direct proofs for more general extensions are given for each class by exploiting the structural properties of G3 sequent calculi; for five of the seven classes the results are strengthened to height-preserving statements, and it is further shown that the constructive and minimal proofs are identical in structure to the classical proof from which they are obtained.

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References

  1. Ishihara H.: A note on the Gödel-Gentzen translation. Math. Log. Q. 46, 135–137 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ishihara H.: Some conservative extension results on classical and intuitionistic sequent calculi. In: Berger, U. et al. (eds.) Logic, Construction, Computation, pp. 289–304. Ontos Verlag, Heusenstamm (2013)

    Google Scholar 

  3. Nadathur G.: Correspondence between classical, intuitionistic and uniform provability. Theoret. Comput. Sci. 232, 273–298 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Negri S.: Contraction-free sequent calculi for geometric theories, with an application to Barr’s theorem. Arch. Math. Log. 42, 389–401 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Negri, S.: Proof analysis beyond geometric theories: from rule systems to systems of rules. J. Log. Comput. doi:10.1093/logcom/exu037 (2014)

  6. Negri S., von Plato J.: Structural Proof Theory. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  7. Negri S., von Plato J.: Proof Analysis. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  8. Orevkov, V.P.: On Glivenko sequent classes. Proc. Steklov Inst. Math., vol. 98, pp. 147–173. (Translated from the Russian original) V. P. Orevkov, Glivenko’s sequence classes, Logical and logical-mathematical calculus, Part I. Trudy Mat. Inst. Steklov, vol. 98, pp. 131–154 (1968)

  9. Schwichtenberg H., Senjak C.: Minimal from classical proofs. Ann. Pure Appl. Log. 164, 740–748 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Troelstra A., Schwichtenberg H.: Basic Proof Theory. 2nd edn. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

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  1. Department of Philosophy, University of Helsinki, Helsinki, Finland

    Sara Negri

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  1. Sara Negri
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Correspondence to Sara Negri.

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Negri, S. Glivenko sequent classes in the light of structural proof theory. Arch. Math. Logic 55, 461–473 (2016). https://doi.org/10.1007/s00153-016-0474-y

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  • DOI: https://doi.org/10.1007/s00153-016-0474-y

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